How Long Would It Take the Whole World to Do the Ice Bucket Challenge?

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How Long Would It Take the Whole World to Do the Ice Bucket Challenge?

Rhett Allain

Even though I am starting to get tired of the ALS Ice Bucket Challenge, you have to admit that it's a brilliant strategy. According to Wikipedia, the challenge works like this:

A nominated person pours a bucket of ice water on their head.

If the person does not wish to do the ice bucket, that person can instead donate to the ALS Association.

The nominated person is often required to do the challenge (or donate) within 24 hours.

Afterwards, the ice bucketed person nominates 3 other people to do the same thing.

It's sort of like a virus in that the more people do the challenge, the more people are nominated. So, how long would it take before the whole world has completed the Ice Bucket Challenge? Let's estimate this.

Ice Challenge Model 1
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In this first model, I will make the following assumptions.

Someone does the first Ice Bucket Challenge.

This person then picks three other people to do the challenge.

These three people then pick 3 people for the challenge.

Each new generation does the challenge 2 days after the announcement (all at the same time) and no one declines.

Everyone that is nominated hasn't already participated in the IBC (Ice Bucket Challenge).

This keeps going until the whole world (7 billion people) have completed the challenge. So, how long would this take? It probably wouldn't be too hard to make a mathematical model for this problem - but I'm just going to do it computationally in python. Actually, this is super simple. All I need to do is to make a loop. If I start the loop with n1 people that have done the challenge, then after the loop, the total number of people that have completed IBC would be:

Yes, I could just write this as 4n1 - but I like it this way for now. Next, I just keeping doing this calculation until I get to 7 billion people. It's that easy.

Now for the data. Here is the plot of the number of IBCers as a function of day.

Notice that the vertical axis is a log-scale (just to be clear). From this you can also see that in just under 35 days, THE WHOLE WORLD will have completed the Ice Bucket Challenge. Actually, in 35 days more than the whole world would have done the challenge - that must include Mars.

Why is this a straight line? By saying that each step is a multiple of the previous step, I have made an exponential function. When you take the log of an exponential function, you get a straight line.

Ice Challenge Model 2 - A Little More Realistic
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Clearly there were some problems with the previous model. Let me make some changes.

When someone nominates a new human, there is a chance that human has already completed the IBC.

Suppose the probability of picking a new (virgin) challenger is dependent on the number IBC finishers compared to the total population.

So, for each generation the probability of picking new humans would be:

With this, when the challenge first starts out the probability of finding someone new would be 100% (since no one else has done it). When most of the population has already done the challenge, the probability of finding someone new is very low.

Ok, let's model this. In the best case, I would make a list of people. For each IBC, I would use a random function to determine which people in the human get the new challenge. Then I would see if those people had already done the challenge. But I'm not going to do this. Why not? Because I don't want to deal with a list with 7 billion items.

Instead, I am going to cheat. Let me give an example. Suppose that there are 100 people on the Earth and 80 of them have done the IBC. When they go to pick the new people, there is an 80% chance that those people have already completed the challenge. This means that only 20% of them will actually do it. Instead of using a random function to figure out who gets picked, I am just going to say 20% actually get picked. This might not be such a bad assumption (even though it isn't correct). Since I am dealing with massive numbers - I am going to say on average 20% would move on to do the challenge.

Now for the plot with this new IBC model 2 along with model 1.

The new model looks just like the old model (mostly). Why? Well, let's look at IBC day 29. On this day, about 268 million people have completed the challenge. That still just about leaves 7 billion people that have NOT done the challenge. So, the probability adjustment for model 2 is insignificant. Only on the very last round do you see a difference between the two models. But by then, it's too late. The last round still gets just about the whole world covered in ice water.

__Update (8/20/14): __As pointed out by a reader (HT Lee-Jon Ball), I made a mistake. In my calculation, I assumed that every two days everyone that has completed the ice bucket challenge will nominate 3 people. This is wrong. Only the people from the previous round would nominate 3 people. This will slightly change the date for the whole world to do the challenge.